An Alexandrov topology is a topology in which an arbitrary intersection of open sets is open. The wikipedia article has some characterizations in terms of category theory:
- Finite inclusion map: The inclusion maps $f_i:X_i\to X$ of the finite subspaces of $X$ form a final sink.
- Finite generation: $X$ is finitely generated i.e. it is in the final hull of the finite spaces.
What does final sink and final hull mean? Can you suggest any references about this topic?
(And does the second bullet mean the final hull of its final subspaces, or the final hull of finite spaces in general, whatever final hull is supposed to mean?)
A sink is an indexed set of morphisms that all share the same codomain (but the domain of each morphism may be different).
We call a sink in $\mathrm{Top}$ final if the shared codomain of the sink carries the finest topology (on the underlying set of the codomain) that makes all the morphisms of the sink continuous.
The source/sink terminology is usually associated with the work of Jiří Adámek [1]. Indeed, I suspect that he came up with these two characterizations of Alexandrov spaces.
Since you seek references, I'd suggest consulting Adámek's textbooks directly: the freely available Abstract and Concrete Categories, along with the higher-level Locally Presentable and Accessible Categories should cover sources, sinks, their initiality/finality, as well as finite generation.
[1] But probably originates from a category theory textbook of Horst Herrlich and George E. Strecker, as pointed out by Tyrone in the comments.